Graduate Abstract Algebra I
Rutgers MATH 551, Fall 2019.
daniel.krashen+alg1@gmail.com
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Lecture 1
: Introduction to algebraic structures via universal algebra
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Lecture 2
: Monoid-oids (aka Categories), some general remarks about groups
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Lecture 3
: The orbit-stabilizer theorem and the class equation
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Lecture 4
: The class equation!
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Lecture 5
: Putting groups together from their pieces
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Lecture 6
: Putting groups back together -- group extensions
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Lecture 7
: An accounting of structures, Nilpotent groups, part 1
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Lecture 8
: Free groups, part 1
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Lecture 9
: Free groups, part 2: universal properties and adjunctions
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Lecture 10
: Nilpotent groups, part 2; Solvable groups
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Lecture 11
: Rings: definitions and examples
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Lecture 12
: Basic properties of rings and ideals; fractions and localization, part 1
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Lecture 13
: Fractions and localization, part 1; Chinese remainder theorem; Principal ideal domains
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Lecture 14
: Unique factorization domains and Gauss' Lemma
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Lecture 15
: The Hilbert basis theorem; Introduction to modules
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Lecture 16
: More modules; tensor products, part 1
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Lecture 17
: Tensor products, part 2
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Lecture 18
: Tensor products, part 3; bilinear forms; intro to vector spaces
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Lecture 19
: Vector space notions: duals, matrices, Kroneker products, tensor algebras
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Lecture 20
: Symmetric and Exterior algebras. Start of modules over a PID
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Lecture 21
: The structure of modules over a PID
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Lecture 22
: Examples and application of the structure theory for modules over a PID
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Lecture 23
: Jordan-Holder for X-groups. Setting the stage for the Yoneda Lemma
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Lecture 24
: Some categorical notions and the Yoneda Lemma. Introduction to representations of finite groups.
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Lecture 25
: Representations of finite groups, part 2.
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Lecture 26
: Representations of finite groups, part 3; Burnside's Theorem.
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